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arxiv: 2506.08307 · v3 · pith:OVL6FRUPnew · submitted 2025-06-10 · 🧮 math.CV

Monogenic functions over real alternative *-algebras: the several hypercomplex variables case

Zhenghua Xu , Chao Ding , Haiyan Wang This is my paper

Pith reviewed 2026-05-22 01:36 UTC · model grok-4.3

classification 🧮 math.CV
keywords monogenic functionsalternative *-algebrashypercomplex variablesBochner-Martinelli formulaPlemelj-Sokhotski formulaHartogs extension theoremseveral variables
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The pith

Monogenic functions of several hypercomplex variables over real alternative *-algebras satisfy the Bochner-Martinelli formula, the Plemelj-Sokhotski formula, and the Hartogs extension theorem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the single-variable notion of monogenic functions to the several-variable case inside real alternative *-algebras. This produces a common setting that contains quaternionic, octonionic, and Clifford analysis as special instances. Within the new setting the authors prove that the classical integral representation and jump formulas, together with the Hartogs extension property, continue to hold. A reader cares because the extension keeps the analytic toolkit intact while moving to a strictly larger class of algebras.

Core claim

In the setting of several hypercomplex variables over real alternative *-algebras, monogenic functions admit the Bochner-Martinelli integral formula, satisfy the Plemelj-Sokhotski jump formula, and obey the Hartogs extension theorem, thereby extending the single-variable monogenic theory to the multi-variable case.

What carries the argument

Monogenic functions of several hypercomplex variables, defined via the generalized Cauchy-Riemann operator over real alternative *-algebras; this definition permits the standard integral and extension arguments to transfer from the single-variable theory.

Load-bearing premise

The single-variable monogenic theory over real alternative *-algebras extends without obstruction to the several-variable case.

What would settle it

An explicit counterexample in two variables over the octonions where the Bochner-Martinelli formula fails to reproduce the function inside a bounded domain would falsify the central claim.

read the original abstract

The notion of monogenic (or regular) functions, which is a correspondence of holomorphic functions, has been studied extensively in hypercomplex analysis, including quaternionic, octonionic, and Clifford analysis. Recently, the concept of monogenic functions over real alternative $\ast$-algebras has been introduced to unify several classical monogenic functions theories. In this paper, we initiate the study of monogenic functions of several hypercomplex variables over real alternative $\ast$-algebras, which naturally extends the theory of several complex variables to a very general setting. In this new setting, we develop some fundamental properties, such as Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends the recently introduced notion of monogenic functions over real alternative *-algebras from the single-variable to the several hypercomplex variables setting. It develops the corresponding Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem, providing explicit constructions and proofs that remain within the alternativity axioms.

Significance. If the derivations hold, the work supplies a broad unifying framework that generalizes classical results from several complex variables, quaternionic analysis, octonionic analysis, and Clifford analysis to a single setting of real alternative *-algebras. The explicit kernel identities and Stokes-type arguments that avoid hidden associativity assumptions constitute a clear technical strength.

minor comments (3)
  1. [§2] §2 (Definition of several-variable monogenicity): the precise statement of how the left and right monogenic conditions are imposed simultaneously on the several variables could be stated as a numbered definition for easier reference in later proofs.
  2. [§4] §4 (Bochner-Martinelli formula): the kernel is written with a summation over multi-indices; a short remark clarifying that the alternativity is used only to verify the required cancellation in the exterior derivative would improve readability.
  3. The paper cites the single-variable theory but does not include a self-contained one-paragraph recap of the key algebraic identities that are reused; adding this would make the several-variable extension more self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We appreciate the recognition that the work provides a unifying framework generalizing results from several complex variables, quaternionic, octonionic, and Clifford analysis while remaining within the alternativity axioms.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper extends the single-variable monogenic function definition over real alternative *-algebras (cited as recently introduced) to the several hypercomplex variables setting and then supplies explicit constructions and proofs for the Bochner-Martinelli formula, Plemelj-Sokhotski formula, and Hartogs extension theorem. These steps rely on the alternativity axioms, kernel identities, and Stokes-type arguments that remain internal to the new framework without reducing any claimed result to a fitted parameter, self-referential equation, or unverified self-citation chain. The central claims therefore rest on independent mathematical content rather than circular reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the definition of monogenic functions from the authors' recent prior work on the single-variable case over real alternative *-algebras. No new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Monogenic functions over real alternative *-algebras are well-defined in the single-variable case (prior work).
    All new several-variable results are built on this definition.

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Works this paper leans on

35 extracted references · 35 canonical work pages · 3 internal anchors

  1. [1]

    W. W. Adams, C. A. Berenstein, P. Loustaunau, I. Sabadini, D. C. Struppa, Regular functions of several quaternionic variables and the Cauchy-Fueter com- plex, J. Geom. Anal. 9 (1999), no. 1, 1-15

    work page 1999

  2. [2]

    Bochner, Analytic and meromorphic continuation by means of Green’s for- mula, Ann

    S. Bochner, Analytic and meromorphic continuation by means of Green’s for- mula, Ann. of Math. (2) 44 (1943), 652-673. 18

    work page 1943

  3. [3]

    Bory Reyes, A

    J. Bory Reyes, A. Guzm´ an Ad´ an, F. Sommen,Bochner-Martinelli formula in superspace, Math. Methods Appl. Sci. 41 (2018), no. 18, 9449-9476

    work page 2018

  4. [4]

    Brackx, B

    F. Brackx, B. De Knock, H. De Schepper, F. Sommen, On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 395-416

    work page 2009

  5. [5]

    Brackx, R

    F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Research Notes in Math- ematics, Vol. 76, Pitman, Boston, 1982

    work page 1982

  6. [6]

    Brackx, W

    F. Brackx, W. Pincket, A Bochner-Martinelli formula for the biregular functions of Clifford analysis , Complex Variables Theory Appl. 4 (1984), no. 1, 39-48

    work page 1984

  7. [7]

    Colombo, I

    F. Colombo, I. Sabadini, F. Sommen, D. C. Struppa, Analysis of Dirac sys- tems and computational algebra , Progress in Mathematical Physics, Vol. 39, Birkh¨ auser Boston, 2004

    work page 2004

  8. [8]

    Delanghe, F

    R. Delanghe, F. Sommen, V. Souˇ cek, Clifford algebra and spinor-valued func- tions. A function theory for the Dirac operator , Mathematics and its Applica- tions, Vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992

    work page 1992

  9. [9]

    Dentoni, M

    P. Dentoni, M. Sce, Funzioni regolari nell’algebra di Cayley , Rend. Sem. Mat. Univ. Padova 50 (1973), 251-267

    work page 1973

  10. [10]

    A unified theory of regular functions of a hypercomplex variable

    R. Ghiloni, C. Stoppato, A unified theory of regular functions of a hypercomplex variable, arXiv:2408.01523, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  11. [11]

    G¨ urlebeck, K

    K. G¨ urlebeck, K. Habetha, W. Spr¨ oßig,Holomorphic functions in the plane and n-dimensional space, Birkh¨ auser Verlag, Basel, 2008

    work page 2008

  12. [12]

    Q. Huo, G. Ren, Z. Xu, Monogenic functions over real alternative ∗-algebras: fundamental results and applications , arXiv:2504.01359, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  13. [13]

    A. M. Kytmanov, The Bochner-Martinelli integral and its applications , Birkh¨ auser Verlag, Basel, 1995

    work page 1995

  14. [14]

    X. Li, L. Peng, The Cauchy integral formulas on the octonions , Bull. Belg. Math. Soc. Simon Stevin 9 (2002), no. 1, 47-64

    work page 2002

  15. [15]

    Y. Li, G. Ren, Monogenic Cauchy implies holomorphic Bochner-Martinelli , Adv. Appl. Clifford Algebr. 32 (2022), no. 3, Paper No. 34, 19 pp

    work page 2022

  16. [16]

    Lin, The boundary behavior of Cauchy type on a closed pieeewise smooth manifold, (Chinese) Acta Math

    L. Lin, The boundary behavior of Cauchy type on a closed pieeewise smooth manifold, (Chinese) Acta Math. Sinica, 31 (1988), 547-557

    work page 1988

  17. [17]

    L. Lin, C. Qiu, The singular integral equation on a closed piecewise smooth manifold in Cn, Integral Equations Operator Theory 44 (2002), no. 3, 337-358

    work page 2002

  18. [18]

    L. Lin, C. Qiu, The Cauchy boundary value problems on closed piecewise smooth manifolds in Cn, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 989-998

    work page 2004

  19. [19]

    Q. Lu, T. Zhong, An extension of the Privalov theorem , (Chinese) Acta Math. Sinica, 7 (1957), 144-165. 19

    work page 1957

  20. [20]

    Marmolejo-Olea, M

    E. Marmolejo-Olea, M. Mitrea, Harmonic analysis for general first order dif- ferential operators in Lipschitz domains , Clifford algebras, 91-114, Prog. Math. Phys., 34, Birkh¨ auser Boston, Boston, MA, 2004

    work page 2004

  21. [21]

    Martinelli, Alcuni teoremi integrali per le funzioni analitiche di pi` u variabili complesse

    E. Martinelli, Alcuni teoremi integrali per le funzioni analitiche di pi` u variabili complesse. Mem. della Reale Accad. d’Italia 9, (1938), 269-283

    work page 1938

  22. [22]

    Okubo, Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics, 2

    S. Okubo, Introduction to octonion and other non-associative algebras in physics, Montroll Memorial Lecture Series in Mathematical Physics, 2. Cam- bridge University Press, Cambridge, 1995

    work page 1995

  23. [23]

    Perotti, Cauchy-Riemann operators and local slice analysis over real alter- native algebras, J

    A. Perotti, Cauchy-Riemann operators and local slice analysis over real alter- native algebras, J. Math. Anal. Appl. 516 (2022), no. 1, Paper No. 126480, 34 pp

    work page 2022

  24. [24]

    Pertici, Regular functions of several quaternionic variables , (Italian) Ann

    D. Pertici, Regular functions of several quaternionic variables , (Italian) Ann. Mat. Pura Appl. (4) 151 (1988), 39-65

    work page 1988

  25. [25]

    T. Qian, T. Zhong, The differential integral equations on smooth closed ori- entable manifolds, Acta Math. Sci. Ser. B (Engl. Ed.) 21 (2001), no. 1, 1-8

    work page 2001

  26. [26]

    R. M. Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, 108. Springer-Verlag, New York, 1986

    work page 1986

  27. [27]

    G. Ren, Y. Shi, W. Wang, The tangential k-Cauchy-Fueter operator and k-CF functions over the Heisenberg group , Adv. Appl. Clifford Algebr. 30 (2020), no. 2, Paper No. 20, 19 pp

    work page 2020

  28. [28]

    G. Ren, H. Wang, Theory of several paravector variables: Bochner-Martinelli formula and Hartogs theorem , Sci. China Math. 57 (2014), no. 11, 2347-2360

    work page 2014

  29. [29]

    Rocha-Ch´ avez, M

    R. Rocha-Ch´ avez, M. Shapiro, F. Sommen,Integral theorems for functions and differential forms in Cm, Chapman and Hall/CRC, Boca Raton, FL, 2002

    work page 2002

  30. [30]

    R. D. Schafer, An introduction to nonassociative algebras , Pure and Applied Mathematics, Vol. 22. Academic Press, New York-London, 1966

    work page 1966

  31. [31]

    Sommen, Martinelli-Bochner type formulae in complex Clifford analysis , Z

    F. Sommen, Martinelli-Bochner type formulae in complex Clifford analysis , Z. Anal. Anwendungen 6 (1987), no. 1, 75-82

    work page 1987

  32. [32]

    H. Wang, G. Ren, Octonion analysis of several variables , Commun. Math. Stat. 2 (2014), no. 2, 163-185

    work page 2014

  33. [33]

    H. Wang, G. Ren, Bochner-Martinelli formula for k-Cauchy-Fueter operator, J. Geom. Phys. 84 (2014), 43-54

    work page 2014

  34. [34]

    Wang, The k-Cauchy-Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic k-regular functions, J

    W. Wang, The k-Cauchy-Fueter complex, Penrose transformation and Hartogs phenomenon for quaternionic k-regular functions, J. Geom. Phys. 60 (2010), no. 3, 513-530

    work page 2010

  35. [35]

    Z. Xu, I. Sabadini, Generalized partial-slice monogenic functions: the octonionic case, arXiv:2503.12409, 2025. 20

    work page internal anchor Pith review Pith/arXiv arXiv 2025